Recently, a number of interesting relations have been discovered between generalised Pauli/Dirac groups and certain finite geometries. Here, we succeeded in finding a general unifying framework for all these relations. We introduce gradually necessary and sufficient conditions to be met in order to carry out the following programme: Given a group $vG$, we first construct vector spaces over $GF(p)$, $p$ a prime, by factorising $vG$ over appropriate normal subgroups. Then, by expressing $GF(p)$ in terms of the commutator subgroup of $vG$, we construct alternating bilinear forms, which reflect whether or not two elements of $vG$ commute. Restricting to $p=2$, we search for ``refinements in terms of quadratic forms, which capture the fact whether or not the order of an element of $vG$ is $leq 2$. Such factor-group-generated vector spaces admit a natural reinterpretation in the language of symplectic and orthogonal polar spaces, where each point becomes a ``condensation of several distinct elements of $vG$. Finally, several well-known physical examples (single- and two-qubit Pauli groups, both the real and complex case) are worked out in detail to illustrate the fine traits of the formalism.
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